In any of several fields of study that treat the use of signs—for example, inlinguistics,logic,mathematics,semantics,semiotics, andphilosophy of language—anintension is anyproperty orqualityconnoted by aword,phrase, or another symbol.[1] In the case of a word, the word'sdefinition often implies an intension. For instance, the intensions of the wordplant include properties such as "being composed ofcellulose (not always true)", "alive", and "organism", among others. Acomprehension is the collection of all such intensions.
The meaning of a word can be thought of as the bond between theidea the word means and thephysical form of the word. Swiss linguistFerdinand de Saussure (1857–1913) contrasts three concepts:
Without intension of some sort, a word has no meaning.[2] For instance, the termsrantans orbrillig have no intension and hence no meaning. Such terms may be suggestive, but a term can besuggestive without being meaningful. For instance,ran tan is an archaic onomatopoeia for chaotic noise or din and may suggest to English speakers a din or meaningless noise, andbrillig though made up byLewis Carroll may be suggestive of 'brilliant' or 'frigid'. Such terms, it may be argued, are always intensional since they connote the property 'meaningless term', but this is only an apparent paradox and does not constitute a counterexample to the claim that without intension a word has no meaning. Part of its intension is that it has noextension. Intension is analogous to the signified in the Saussurean system, extension to the referent.
In philosophical arguments aboutdualism versusmonism, it is noted that thoughts have intensionality and physical objects do not (S. E. Palmer, 1999), but rather have extension in space and time.
A statement-form is simply a form obtained by putting blanks into a sentence where one or more expressions with extensions occur—for instance, "The quick brown ___ jumped over the lazy ___'s back." An instance of the form is a statement obtained by filling the blanks in.
Anintensional statement-form is a statement-form with at least one instance such that substituting co-extensive expressions into it does not always preservelogical value. Anintensional statement is a statement that is an instance of an intensional statement-form. Here co-extensive expressions are expressions with the sameextension.[3]
That is, a statement-form is intensional if it has, as one of its instances, a statement for which there are two co-extensive expressions (in the relevant language) such that one of them occurs in the statement, and if the other one is put in its place (uniformly, so that it replaces the former expression wherever it occurs in the statement), the result is a (different) statement with a different logical value. An intensional statement, then, is an instance of such a form; it has the same form as a statement in which substitution of co-extensive terms fails to preserve logical value.
The first example has a different logical value if the term "Mark Twain" is replaced with the co-extensive term "The author ofCorn-pone Opinions", since not everyone who has readHuckleberry Finn knows that the same author also wroteCorn-pone Opinions.
The second example has a different logical value if the term "stargazing" is replaced with the co-extensive term "looking at luminousspheroids ofplasma held together byself-gravity", since Aristotle would not have been aware of this definition of the term "star", and therefore would not have used it in a remark.
The intensional statements above feature expressions like "knows", "possible", and "pleased". Such expressions always, or nearly always, produce intensional statements when added (in some intelligible manner) to an extensional statement, and thus they (or more complex expressions like "It is possible that") are sometimes calledintensional operators. A large class of intensional statements, but by no means all, can be spotted from the fact that they contain intensional operators.
Anextensional statement is a non-intensional statement. Substitution of co-extensive expressions into it always preserves logical value. A language is intensional if it contains intensional statements, and extensional otherwise. All natural languages are intensional.[4] The only extensional languages are artificially constructed languages used inmathematical logic or for other special purposes and small fragments of natural languages.
Note that if "Samuel Clemens" is put into (1) in place of "Mark Twain", the result is as true as the original statement. It should be clear that no matter what is put for "Mark Twain", so long as it is a singular term picking out the same man, the statement remains true. Likewise, we can put in place of thepredicate any other predicate belonging to Mark Twain and only to Mark Twain, without changing the logical value.
For (2), the term "stargazing" can now be substituted with "looking at luminous spheroids of plasma held together by self-gravity", since Aristotle personally being aware of the two terms being co-extensive is no longer relevant to the logical value of the sentence.